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The question in your image asks to find the percent abundance of two isotopes given their individual masses and the average atomic mass of the element.

### a. The general equation for calculating the atomic mass of an element with two isotopes is:

\[
\text{Atomic mass} = (m_1 \cdot f_1) + (m_2 \cdot f_2)
\]

Where:
- \( m_1 \) is the mass of the first isotope.
- \( m_2 \) is the mass of the second isotope.
- \( f_1 \) is the fractional abundance of the first isotope.
- \( f_2 \) is the fractional abundance of the second isotope.
- \( f_1 + f_2 = 1 \) (since the total abundance must be 100%).

### Given values:
- \( m_1 = 12.2959 \, \text{amu} \)
- \( m_2 = 15.9921 \, \text{amu} \)
- Atomic mass \( = 13.9362 \, \text{amu} \)

### b & c. Now, we will solve for the percent abundances of both isotopes.

Let \( f_1 \) be the fractional abundance of isotope 1, and \( f_2 = 1 - f_1 \) be the fractional abundance of isotope 2.

Substitute the values into the atomic mass equation:

\[
13.9362 = (12.2959 \cdot f_1) + (15.9921 \cdot (1 - f_1))
\]

Now, expand the terms:

\[
13.9362 = 12.2959 \cdot f_1 + 15.9921 - 15.9921 \cdot f_1
\]

Simplify:

\[
13.9362 = (12.2959 - 15.9921) \cdot f_1 + 15.9921
\]
\[
13.9362 = -3.6962 \cdot f_1 + 15.9921
\]

Now, isolate \( f_1 \):

\[
13.9362 - 15.9921 = -3.6962 \cdot f_1
\]
\[
-2.0559 = -3.6962 \cdot f_1
\]
\[
f_1 = \frac{-2.0559}{-3.6962} \approx 0.556
\]

Thus, the fractional abundance of isotope 1 is approximately 0.556, or 55.6%.

Since \( f_2 = 1 - f_1 \):

\[
f_2 = 1 - 0.556 = 0.444
\]

The fractional abundance of isotope 2 is approximately 0.444, or 44.4%.

### Final Answers:
- Percentage abundance of isotope 1: **55.6%**
- Percentage abundance of isotope 2: **44.4%**

Would you like more details on this? Here are some additional related questions:

1. How do you calculate the atomic mass when more than two isotopes are involved?
2. What factors affect the natural abundance of isotopes?
3. How is the atomic mass used in determining the stability of an isotope?
4. What is the relationship between isotope abundance and atomic mass on the periodic table?
5. How are isotopes detected experimentally in a laboratory?

**Tip**: Always ensure to double-check for correct rounding when calculating percent abundances to match the significant figures in the problem.

A scientist discovered a new element that has two naturally occurring isotopes. The first isotope has a mass of 12.2959 amu and the second isotope has a mass of 15.9921 amu. If the element has an atomic mass of 13.9362 amu, what is the percent abundance of each isotope?

Solve the problem of finding the percent abundance of two isotopes with masses 12.2959 amu and 15.9921 amu given an atomic mass of 13.9362 amu. This question involves weighted averages and linear equations, ideal for high school students.

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